A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

Anderson, David F.; Blom, Joke; Mandjes, Michel; Thorsdottir, Halldora; Turck, Koen De

doi:10.1007/s11009-014-9405-8

Cited by 31 publications

(64 citation statements)

References 28 publications

(38 reference statements)

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“…When inflating the arrival rates by a factor N , and speeding up the background process by a factor N α (for some α > 0), in e.g. ( Anderson, Blom, Mandjes, Thorsdottir, and De Turck, 2014;Blom, De Turck, and Mandjes, 2015;Blom, De Turck, and Mandjes, 2016 ) it has been proven that the (transient as well as stationary) number of jobs present in the system is, after centering and normalizing, asymptotically Normally distributed. An interesting dichotomy was identified, in that the regimes α < 1 and α > 1 lead to qualitatively different asymptotics.…”

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confidence: 99%

Refined large deviations asymptotics for Markov-modulated infinite-server systems

Blom

Turck

Mandjes

2017

European Journal of Operational Research

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a b s t r a c tMany networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers' arrival rates and service rates are modulated by a Markovian background process; additionally, there are infinitely many servers (and consequently the resulting model is often used as a proxy for the corresponding many-server model). The Markov-modulated infinite-server model hardly allows any explicit analysis, apart from results in terms of systems of (ordinary or partial) differential equations for the underlying probability generating functions, and recursions to obtain all moments. As a consequence, recent research efforts have pursued an asymptotic analysis in various limiting regimes, notably the central-limit regime (describing fluctuations around the average behavior) and the largedeviations regime (focusing on rare events). Many of these results use the property that the number of customers in the system obeys a Poisson distribution with a random parameter. The objective of this paper is to develop techniques to accurately approximate tail probabilities in the large-deviations regime. We consider the scaling in which the arrival rates are inflated by a factor N , and we are interested in the probability that the number of customers exceeds a given level Na . Where earlier contributions focused on so-called logarithmic asymptotics of this exceedance probability (which are inherently imprecise), the present paper improves upon those results in that exact asymptotics are established. These are found in two steps: first the distribution of the random parameter of the Poisson distribution is characterized, and then this knowledge is used to identify the exact asymptotics. The paper is concluded by a set of numerical experiments, in which the accuracy of the asymptotic results is assessed.

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confidence: 99%

Refined large deviations asymptotics for Markov-modulated infinite-server systems

Blom

Turck

Mandjes

2017

European Journal of Operational Research

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“…10.3). This limiting behavior continues to hold in case the queueing process is modulated by a Markovian background process, see Anderson et al (2016 Stochastic Systems, 2017, vol. 7, no.…”

Section: Related Literaturementioning

confidence: 83%

A Blood Bank Model with Perishable Blood and Demand Impatience

Bar‐Lev

Boxma²,

Mathijsen³

et al. 2017

Stochastic Systems

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Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditionsThis article may be used only for the purposes of research, teaching, and/or private study. Commercial use or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher approval, unless otherwise noted. For more information, contact permissions@informs.org.The Publisher does not warrant or guarantee the article's accuracy, completeness, merchantability, fitness for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. Copyright © 2017, The Author(s) Please scroll down for article-it is on subsequent pagesINFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. Abstract. We consider a stochastic model for a blood bank, in which amounts of blood are offered and demanded according to independent compound Poisson processes. Blood is perishable, i.e., blood can only be kept in storage for a limited amount of time. Furthermore, demand for blood is impatient, i.e., a demand for blood may be canceled if it cannot be satisfied soon enough. For a range of perishability functions and demand impatience functions, we derive the steady-state distributions of the amount of blood kept in storage, and of the amount of demand for blood (at any point in time, at most one of these quantities is positive). Under certain conditions we also obtain the fluid and diffusion limits of the blood inventory process, showing in particular that the diffusion limit process is an Ornstein-Uhlenbeck process.

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“…In the following, we assume that the service requirements are exponentially distributed with rate μ, and we point out how it can be generalized to a general distribution in Remark 3.6 below. We follow the standard approach to derive the FCLT for infinite-server queueing systems; we mimic the argumentation used in, for example [1,14]. As the proof has a relatively large number of standard elements, we restrict ourselves to the most important steps.…”

Section: Asymptotic Analysismentioning

confidence: 99%

“…Following the reasoning of [1], assuming that N n (0)/n ⇒ ρ(0) (where '⇒' denotes weak convergence), N n (t)/n converges almost surely to the solution of…”

Section: Asymptotic Analysismentioning

confidence: 99%

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Networks of $$\cdot /G/\infty $$ · / G / ∞ queues with shot-noise-driven arrival intensities

2017

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We study infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the arrival rate is given by a shot-noise process. A shot-noise rate emerges naturally in cases where the arrival rate tends to exhibit sudden increases (or shots) at random epochs, after which the rate is inclined to revert to lower values. Exponential decay of the shot noise is assumed, so that the queueing systems are amenable to analysis. In particular, we perform transient analysis on the number of jobs in the queue jointly with the value of the driving shot-noise process. Additionally, we derive heavy-traffic asymptotics for the number of jobs in the system by using a linear scaling of the shot intensity. First we focus on a onedimensional setting in which there is a single infinite-server queue, which we then extend to a network setting.

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A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

Cited by 31 publications

References 28 publications

Refined large deviations asymptotics for Markov-modulated infinite-server systems

Refined large deviations asymptotics for Markov-modulated infinite-server systems

A Blood Bank Model with Perishable Blood and Demand Impatience

Networks of $$\cdot /G/\infty $$ · / G / ∞ queues with shot-noise-driven arrival intensities

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